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Journal of Noncommutative Geometry

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Volume 14, Issue 2, 2020, pp. 773–788
DOI: 10.4171/JNCG/379

Published online: 2020-07-21

Embedding of the derived Brauer group into the secondary $K$-theory ring

Gonçalo Tabuada[1]

(1) Massachusetts Institute of Technology, Cambridge, USA and Universidade NOVA da Lisboa, Portugal

In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree $\geq 4$ as well as in the case of Mumford’s (famous) singular surface.

Keywords: Derived Brauer group, SecondaryK-theory, Noncommutative motives, Noncommutative algebraic geometry

Tabuada Gonçalo: Embedding of the derived Brauer group into the secondary $K$-theory ring. J. Noncommut. Geom. 14 (2020), 773-788. doi: 10.4171/JNCG/379